Thursday, 21 August 2014

Putting the "and" in understanding

Hypothetical situation: let's say you've made some poor decisions in life, and through a series of very unlucky events, you've wound up doing an undergraduate degree in physics. Hey, don't feel bad - we all make mistakes. Happens to the best of us. Unfortunately, though, you're kind of stuck here now, and you figure you should probably try to make the best of it - and that means you'll now have to complete an unreasonably large number of assignments (this is how Physicists ensure that students graduate with the Important Skill of Being Able to Solve Assignment Problems). Anyway, you've just been given the latest assignment, and it's a doozy. For all the sense you can make of it, it may as well be written in Greek (and for that matter, about half of it actually is written in Greek). What's a poor physicist to do?

Well, usually the first thing a poor physicist is to do is procrastinate. But what about after that, when you actually want to get around to solving the problem?

Well, if I were such a hypothetical physicist (and man just to re-emphasize would that ever suck) I would probably start by breaking the problem into sub-problems. That is, I would identify parts of the problem that I didn't understand, parts which were prerequisites for figuring out the actual problem, and then try to understand those parts. Naturally because we're talking about physics this will no doubt involve further breaking the sub-problems down into sub-sub-problems, and those into sub-sub-sub-problems, but the basic idea is there. Reduce until you understand. If you don't understand, keep reducing.

This I think is basically the correct approach, and I fully endorse it. But it does lead to an interesting view of problem solving, which I would summarize as follows:

Understanding is an AND function.

What do I mean by that? Well, let's say our hypothetical assignment problem has 20 reasonably distinct sub-problems that you have to figure out. Initially, you understand basically nothing about the problem, and it is little more than an opaque wall of confusion that makes you want to break down into tears (hypothetically). So out of the 20 sub-problems, you are able to solve...precisely zero of them. Now let's say we give an "understanding" value to each of the sub-problems, which for simplicity we'll say can only be 0 (if you haven't solved the sub-problem) or 1 (if you have solved it). I would then claim that your total "understanding function" for the entire problem is simply the product of these individual understanding factors. Solving the whole problem requires solving every single one of the sub-problems, and if even one of the sub-problems remains unsolved, you haven't gotten the solution yet, so your overall "understanding function" simply remains at zero. This is what I mean by an AND function - to get that pesky understanding function up to one, you have to solve sub-problem one and sub-problem two and sub-problem three, and so on and so forth. So until you understand everything, it feels (to you anyway) pretty much as if you understand nothing.

This picture has some strange implications for the psychology of problem solving. Namely, it suggests that while solving a problem, in many cases progress won't feel like progress. You can be doing great work, solving sub-problem after sub-problem, flipping zero after zero into one after one, and yet to you it feels as if you've done nothing. Your overall understanding is still at zero. Basically, whenever you solve some sub-problem [A], you can always say "oh sure, I understand [A] now, but that's trivial - even with [A] solved, I still don't understand [B], [C], [X], [Y] and [Z]. I'm still confused." The goalposts shift to the next sub-problem, and because your sense of progress is tied to your overall confusion level, and because confusion doesn't go away until you solve the entire problem, it seems as if you've accomplished nothing. If you could step back and objectively evaluate your overall progress, you would of course agree that you had in fact accomplished a great deal - you solved a bunch of sub-problems! But you can't do that, and you're stuck in the middle of the situation, so all you know is that you haven't solved the whole problem. Thus, confusion reigns.

But it's even worse than just that. For in the scenario I outlined above, you already know how many sub-problems there are to solve. You know that once you figure out all twenty of the sub-problems, you'll have solved the whole problem. In a real assignment this isn't the case - you're in a state of uncertainty regarding the total number of sub-problems. Maybe all you have to do is solve the next sub-problem...but maybe not. Maybe you actually have to solve like twelve freakin more. Who knows! And that's not even taking into account the possibility of heading down the wrong problem-solving path - maybe you've solved ten sub-problems that you initially decided were important, but then you realize that whoops, no, they're actually just irrelevant to the problem at hand. Back to square one! This adds a whole extra layer of uncertainty on to the problem, which only increases your confusion - and again, it's the alleviation of confusion that feels like progress to you.

And in fact, it's even worse than that. Because in the real world, not only do you not know how many sub-problems you are away from a solution, you don't even know if there is a solution. Assignment problems are specifically chosen by professors to be solvable given your current level of knowledge (er, usually anyway). Unfortunately, the universe doesn't grade on a curve - it might be that the problem you're working on is completely beyond your skill level, or simply can't be solved. So even if you manage to notice that you're solving sub-problems, and even if you're sure you aren't solving the wrong sub-problems, even then you don't know if you're making real progress.

I think this is a large part of what makes doing Real Research (TM) so hard. You have to keep working even when you're confused for extended periods of time, all the while not being sure if you're actually getting anywhere or not. It's like being lost, blind, in a maze, and you won't even know if an exit exists up until the moment you find it. Einstein famously worked on his theory of General Relativity for eight years before finally completing it - eight years! That's a staggering, almost mind-boggling accomplishment. In a literal sense, I probably can't even imagine the dedication that it took. That's why Einstein tops so many people's lists (certainly my own) of the greatest physicists of all time. He was absolutely a genius, no doubt about that. But the thing that really set him apart, the thing that got him General Relativity when it wasn't even on anyone else's radar, was his willingness to persist so long in the face of confusion.

We could all probably learn something from Einstein here, because I don't think this only applies to the case of doing abstract theoretical research. Probably a lot of the problems we face in everyday life have this character. Maybe you're trying to figure out how to stop procrastinating, or quit smoking, or...be a better haberdasher, or something. Whatever. The point is, maybe none of the things that you've tried so far have worked. It could be that you were totally on the right track, and all you needed to do was one or two extra things on top of what you were already trying. But because to you it felt like you weren't making any progress - because of the tyranny of the AND function - you stopped trying, tragically just short of a solution.

So if you find yourself banging your head against the wall in despair, unable to solve a problem that has been plaguing you for months and months - well, take heart.